The nuclear force is the force that keeps protons and neutrons together in atomic nuclei. It is often said to be due to a pion exchange proposed by Hideki Yukawa in 1935. His Nobel prize lecture Meson theory in its developments gives some background: “As pointed out by Wigner1, specific nuclear forces between two nucleons, each of which can be either in the neutron state or the proton state, must have a very short range of the order of 10-13 cm, in order to account for the rapid increase of the binding energy from the deuteron to the alpha particle”. He was referring to what we call the nuclear binding energy curve, and said this indicates that nuclear forces are “saturated”. He went on to say this: “Heisenberg2 suggested that this could be accounted for, if we assumed a force between a neutron and a proton, for instance, due to the exchange of the electron or, more generally, due to the exchange of the electric charge, as in the case of the chemical bond between a hydrogen atom and a proton”. That’s referring to the hydrogen molecular ion, also known as the dihydrogen ion or the dihydrogen cation. There’s plenty of material about that on the internet:
Public domain H2plus image by TonyMath, see Wikipedia Hydrogen molecule plot from Rod Nave’s hyperphysics
See for example Mark Tuckerman’s CHEM UA-127 course where he gives a potential energy plot and refers to σ (sigma) orbitals as being analogous to hydrogen s orbitals. Or see Rod Nave’s hydrogen molecule article which says “the bond in the hydrogen molecule is ‘saturated’ because it cannot accept another bond”.
The Yukawa interaction
Yukawa’s proposal is nowadays known as the Yukawa interaction. His paper was on the interaction of elementary particles. He said: “the transition of a heavy particle from neutron state to proton state is not always accompanied by the emission of light particles, i.e. a neutrino and an electron, but the energy liberated by the transition is taken up sometimes by another heavy particle, which in turn will be transformed from proton state into neutron state”. He was talking about an interaction akin to a beta decay plus an immediate electron capture, but one which didn’t actually feature electrons and neutrinos. He referred instead to a U-field that could be likened to the scalar potential of the electromagnetic field, but which decreased more rapidly with distance. He said “this field should be accompanied by a new sort of quantum, just as the electromagnetic field is accompanied by the photon”. He used the word quantum rather than pion, and the photon is a vector boson whilst a pion isn’t. But his meaning seems clear enough: the neutron emits a π – negative pion to become a proton, and a proton absorbs it to become a neutron:
He said this quantum would have a mass mu = λh/c and that assuming a wavelength λ of 5 x 10-12 cm, the mass would be 2 x 102 times the electron mass. That’s about 100 MeV. He also referred to the inverse transition from a proton to a neutron state, which would require the emission and absorption of a π+ positive pion. He also said such a quantum has never been seen in experiment because in the ordinary nuclear transformation it can’t be emitted into outer space. But it might “have some bearing on the shower produced by cosmic rays”. That was pointing at a testable prediction. Yukawa had scientific evidence in mind. For example he said “the lowest energy state has the spin 1, which is required from experiment”. He was talking about the spin-1 deuteron, where the proton-neutron spins are parallel. That’s unlike the proton-electron spins in hydrogen. In hydrogen the antiparallel state has the lowest energy because “the electron is not spatially displaced from the proton, but encompasses it”.
The Yukawa interaction is still considered approximately right today
See Hideki Yukawa and the meson theory by Laurie Brown for an overview. It dates from 1986, and says Yukawa’s work began to be noticed only in 1937, when a particle whose mass closely fit the requirements of meson theory was detected in cosmic rays. This turned out to be the muon, not the pion, but theoreticians didn’t know that until 1947 when the pion was discovered for real. For more information see chapter 2 of The Experimental Foundations of Particle Physics by Robert Cahn and Gerson Goldhaber. Also see isospin by Nicholas Kemmer, which dates from 1982. It describes how Heisenberg talked of p-spin in 1932 and based his interaction on the hydrogen molecular ion: “binding arises by a quantum superposition of the two states A and B, with the electron shared in what can be described as an exchange of the electron between the two protons”. Kemmer went on to say the range of the force is short and that “thanks to Pauli’s exclusion principle the mechanism does not provide for binding of many hydrogen atoms to give large molecules. There is a saturation effect”. He also said Heisenberg saw these to be just the kind of features wanted in a nuclear interaction. For further reading see the algebra of grand unified theories by John Baez and John Huerta. In their Isospin and SU(2) section they say Heisenberg’s analogy turned out to be poor, based on the faulty notion that the neutron was composed of a proton and an electron. They also say that in 1936 paper on nuclear forces, Bernard Cassen and Edward Condon employed an analogy between isospin and electron spin. They say the strong force, unlike the electromagnetic force, is the same whether the particles involved are protons or neutrons. They say that there must be a mechanism which can convert protons into neutrons and vice versa. They say that any physical process caused by this force should be described by an intertwining operator. They say two nucleons interact by exchanging pions, and “this is the mechanism for the strong force proposed by Yukawa, still considered approximately right today”.
The Yukawa interaction is not adequate
However the Yukawa interaction is not adequate to model the nuclear force. I think Itay Yavin says it rather well in his PHYS4E03 course: “since these early days it also became clear that pion exchange cannot be the whole story for the strong nuclear force between nucleons for various reasons, in fact it is not even close to being the full story”. See the Wikipedia nuclear force article where you can read that the Yukawa potential models a central force that’s always attractive. It’s overly simplistic, akin to a model that only caters for the linear electric force between the electron and the positron. The nuclear force isn’t like that. Instead when you plot it, it resembles the hydrogen molecule plot. Again see the Wikipedia nuclear force article where you can read that “the nuclear force is powerfully attractive between nucleons at distances of about 1 femtometer (fm, or 1.0 × 10-15 metres) but rapidly decreases to insignificance at distances beyond about 2.5 fm. At distances less than 0.7 fm, the nuclear force becomes repulsive”. The nuclear force is attractive at short range only, and repulsive at even shorter range:
Strong nuclear force plot from the Dux college HSC physics course
You can also read that work on the deuteron by the Rabi group showed that the nuclear force “was not a central force but instead had a tensor character”. This depends on “the interaction between the nucleon spins and the angular momentum of the nucleons”. Note that the electromagnetic field has a tensor character. That’s why we talk about the electromagnetic field tensor – something as simple as a couple of bar magnets experience tensor force. Also note that Isidor Rabi “described and measured” nuclear magnetic resonance in 1938 and received the Nobel prize for it in 1946. In addition, despite what you can read on some authoritative-looking websites, pion exchange is not some kind of ping pong. There are no actual pions flying back and forth between the proton and the neutron. Pion exchange can be likened to the electromagnetic interaction, but hydrogen atoms don’t twinkle and magnets don’t shine.
They only exist in the mathematics of the model
The virtual photons that are said to mediate the electromagnetic interaction are virtual photons. They aren’t real photons. They only exist in the mathematics of the model. There is an underlying reality, in that the electron and the proton “exchange field” such that the resultant hydrogen atom has no significant electromagnetic field. But there are no actual photons flying back and forth between the proton and the electron. In similar vein there are no actual pions flying back and forth between the proton and the neutron. Moreover the neutron is not 140 MeV heavier than the proton, and we don’t see protons and neutrons turning into neutrons and protons. Then when we do see electron capture and beta decay, pions are never ever produced. Furthermore the pion is said to be a spin-0 spinless particle conforming to the Klein-Gordon equation. Oskar Klein and Walter Gordon came up with that in 1926 to describe relativistic electrons, and it doesn’t cater for spin. It is said that “the Klein–Gordon equation correctly describes the spinless relativistic composite particles, like the pion”. But π+ and π− pions are charged particles, and we know that spin is real and that electromagnetic charge is associated with electromagnetic standing waves going around and around. We also know that magnetic moment is associated with spin, that the neutron magnetic moment is not the same as the proton magnetic moment, and that the pion is said to have no magnetic moment. Then there’s an issue with parity in that pions have a parity of -1, and protons and neutrons have the same +1 parity. In addition according to the standard model, protons and neutrons are not elementary particles, and a pion is not some fundamental boson. There’s a whole host of issues. And yet and the Yukawa interaction is “still considered approximately right today”.
Another thing to note is that Yukawa didn’t mention the neutral pion in his 1935 paper. He said his quantum should have a charge of +e or −e. However on many websites you can see references to neutral pions being exchanged. That’s because some of the issues were recognised in the thirties. See the Nature looking back article on Kemmer. In consecutive letters to Nature in 1938, Nicholas Kemmer and Homi Bhabha “developed Yukawa’s theory further and proposed that the Yukawa particle, in Kemmer’s words, has a charged and an uncharged state; the latter, said Bhabha, could explain the close-range proton-proton interaction”. Kemmer said a theory based on the scalar relativistic Schrodinger explanation couldn’t explain the 1S level of the deuteron relative to the 3S ground state, but “a more satisfactory result can be obtained if one admits a vector wave function for the new particle, such as was used by Proca in a different connexion”. That’s Alexandru Proca who was responsible for the Proca action. And this is why hyperphysics includes a page which refers to the π⁰ neutral pion and says “for a proton to attract a neighbouring proton, it must exchange something with it”. I shall depict it thus:
You can find other articles featuring variations on the theme. There’s a similar depiction on Martin Savage’s one pion exchange lecture. James Carr’s nuclear physics hypertextbook features a proton-proton interaction mediated by a neutral pion. The Wikipedia nuclear force article depicts a proton-neutron interaction mediated by a neutral pion. However the other issues remain, including the lack of proton-proton or neutron-neutron bound states. There are no diprotons, and there are no dineutrons. As James Chadwick said, “each neutron is bound to two protons and each proton to two neutrons”.
Pions were thought to be responsible for beta decay
Another issue is that Yukawa proposed that the pion was responsible for beta decay. He said his quantum could be absorbed by a lightweight particle which in consequence will “rise from a neutrino state of negative energy to an electron state of positive energy, thus an anti–neutrino and an electron are emitted simultaneously from the nucleus”. Yukawa’s proposal was dripping with issues, but in 1938 Bhabha reiterated that idea that the pion was responsible for beta decay. He said this: “in order to explain the β-decay, we must then assume that in certain circumstances the electron may absorb a U-particle by becoming a neutrino or vice versa”. Amazingly neither he nor Yukawa were out on a limb. Yukawa had taken his lead from Dirac’s hole theory, wherein space was supposedly full of negative-energy electrons. In Viśvapriya Mukherji’s short history of the meson theory from 1935 to 1943 you can read that Hans Bethe found the idea of a connection between the nuclear force and beta decay to be so attractive that he was very reluctant to give it up. However we know of no negative energy particles. And of course in the standard model, beta decay is said to be caused by the weak interaction and the Wˉ vector boson, not the pion.
A short history of the meson theory
Mukherji also describes how Yukawa had dealt with a spinless meson but others introduced a spin-1 meson. And a neutral meson to “satisfactorily explain the experimentally-established fact that the same charge-independent force exists between two protons or between two neutrons”. The experiment that is said to have established this “fact” was the scattering of protons by protons by Merle Tuve, Norman Heydenburg, and Lawrence Hafstad in 1936. It refers to an accompanying paper the theory of scattering of protons by protons by Gregory Breit, Edward Condon, and Richard Present. This employs statistics to promote the charge independence hypothesis, saying “the only essential difference in the interactions between like and unlike particles is due to the exclusion principle”. However when you have some qualitative understanding of spinors and charge, when you know that the exclusion principle is associated with repulsion, and when you know that the nuclear force is repulsive at short range, this claim is not at all convincing. It’s even less convincing when you have experience of other scattering experiments which employ non-sequiturs to claim that the electron is a point-particle, or that the proton is composed of an infinity of point-particles. Saying the n-p deuteron is the only stable dinucleon because of the Pauli exclusion principle does not suffice. It isn’t adequately describing the nuclear force, where the short-range repulsion is due to the Pauli exclusion principle. It’s like saying the nuclear force is the same apart from the difference. Hence when the situation was reviewed by Eugene Wigner in 1942, he showed that the then-existing experimental evidence “was still inadequate to make any definite statements about the validity of the charge independence hypothesis”.
A chequered history
Mukherji also tells us how Yukawa’s paper was first mentioned by Robert Oppenheimer and Robert Serber in their note on the nature of cosmic ray particles. They were talking about the muons discovered by Carl Anderson and Seth Neddermeyer at Caltech in 1936. These were thought to be pions, and generally taken as evidence for Yukawa’s theory. Note though that Oppenheimer and Serber were not enamoured. They said this: “the reconciliation of the approximate saturation character of nuclear forces with the apparent equality of like and unlike particle forces and with the magnetic moments of neutron and proton could here too be achieved only by an extreme artificiality. These considerations therefore cannot be regarded as the elements of a correct theory”. Mukherji tells of other difficulties. The theory of Yukawa and Sakata “led to the wrong spin-dependence of the nuclear forces”. Herbert Fröhlich, Walter Heitler and Nick Kemmer were “not ready to accept the equality of the pp force and the np force, nor the strongly attractive character of the pp force”. Then after Kemmer accepted further evidence from Tuve et al, a new theoretical reasoning developed “which considered the neutral meson as the sole conveyor of the nuclear forces (to the exclusion of the charged mesons)”. Then came Meller and Rosenfeld with a mixed meson field theory with two entirely separate meson fields. Then in 1938 Yukawa et al, Bhabha and Kemmer together with Fröhlich and Heitler came to the conclusion that the meson “has a spin one, and obeys the Proca equations”. And as late as 1939 the Gamow-Teller theory of nuclear forces in terms of an electron-positron field “was still being pursued by Critchfield and Teller and also Wigner”. See the December 1952 3rd annual Rochester conference meeting report on charge independence and saturation of nuclear forces. It says this: “In this connection Wigner remarked that the term charge independence is most unfortunate since in fact it has nothing to do with charge”. It’s a chequered history. The participants were groping for a new concept. But they didn’t know what a photon was, or an electron, or how charged particles attract. Or how a magnet works. They didn’t even know what charge was. So they were groping in the dark.
A disastrous history
Charlotte Elster at Ohio University gives some more history in her nuclear force lecture. She says the original Yukawa idea of a scalar field interacting with nucleons was soon extended to vectors and to pseudoscalar and pseudovector fields. She refers to Proca and Kemmer, and describes how in 1951 Mituo Taketani, Seitaro Nakamura, and Muneo Sasaki came up with a proposal to divide the nuclear force into long range, intermediate range, and short range regions. The long-range region was dominated by one-pion exchange, the intermediate region was dominated by a two-pion exchange, and the short range region was dominated by multi-pion, heavy-meson, and other exchanges. Talk about doubling down. Elster says the one-pion exchange became well-established as the long-range part of the nuclear force, but that there were tremendous problems with the two-pion exchange, and that “it was impossible to derive a sufficient spin-orbit force from the 2π exchange”. No small wonder, because the spin-orbit interaction is usually a magnetic interaction. Hence as Ruprecht Machleidt said in his 2014 Sendai nuclear force lecture: ”the first pion period results in a disaster”. That’s why Hans Bethe was still asking what holds the nucleus together in 1953.
There is no meson-exchange model for the nuclear force
There’s more of course. For example in 1960 Gregory Breit suggested heavy vector bosons “to account for the empirically well-established short-ranged spin-orbit force”. Hence on the Wikipedia rho meson article you can read that “along with pions and omega mesons, the rho meson carries the nuclear force within the atomic nucleus”. But when you take a look at more recent material such as the 2007 Riken don’t get too close article, you can see that the issues are still outstanding. You can read that at separations greater than two femtometers the nuclear force is mainly communicated by one-pion exchange, that closer in the exchange consists of multipions and heavy mesons, and that closer still the force is strongly repulsive but the cause “remains an open question”. In the 2009 modern theory of nuclear forces paper by Evgeny Epelbaum et al you can see a plot of the nucleon-nucleon potential. The caption says the longest-range contribution is the one-pion-exchange, the intermediate range attraction is described by two-pion exchanges plus other shorter-ranged contributions, and that at even shorter distances, “the NN interaction is strongly repulsive”. But it doesn’t say why. The bottom line is that Yakuwa wrote his paper in 1935, and here we are 83 years later, and there is still no viable meson-exchange model for the nuclear force.
The residual strong force?
You might think that’s because things moved on in the early sixties with the introduction of the quark model and quantum chromodynamics. After all, you can read that the nuclear force is said to be a by-product of the color force which is due to a gluon exchange between quarks. The Wikipedia strong interaction article is faithful to what particle physicists typically say. It says the nuclear force aka residual strong force is “a minor residuum of the strong force that binds quarks together into protons and neutrons. This same force is much weaker between neutrons and protons, because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (van der Waals forces) are much weaker than the electromagnetic forces that hold electrons in association with the nucleus, forming the atoms”. The color force goes back to 1964. In Harald Fritzsch’s history of QCD you can read that the Ω– omega baryon was in breach of the Pauli exclusion principle and that this was “a great problem for the quark model”.
Sleight of hand
Hence as you can read in Andrew Watson’s book the quantum quark, Oscar Greenberg came up with a fix which would eventually become color charge. The CERN twenty-five years of gluons article says this hypothesis “sounded at first like sleight of hand”. That’s perhaps because the Pauli exclusion principle was never addressed in the context of the quark model, or in the context of the nuclear force. Intramolecular forces have a repulsive component because of the Pauli exclusion principle. Neutron degeneracy pressure has a repulsive component because of the Pauli exclusion principle. That was used to say the deuteron is the only dinucleon, but the short-range repulsion is still there in the deuteron. It just doesn’t fly. Something else that doesn’t fly is the way the residual strong force is still presented as a Yukawa interaction. See the Wikipedia nuclear force article where you can see a gif showing gluon exchange between quarks and pion exchange between nucleons:
CCASA image by Manishearth, see Wikipedia
You can read that “the nuclear force is a residual effect of the more fundamental strong force”. And that “the strong interaction is the attractive force that binds the elementary particles called quarks together to form the nucleons”. Elsewhere in the article you can read that in the light of QCD, the meson-exchange concept is no longer perceived as fundamental, but “continues to represent the best working model for a quantitative NN potential”. If you were to ask a particle physicist if this was true, I’m confident most would say yes. Even though the meson-exchange concept doesn’t work very well. Because the QCD model for a “quantitative NN potential” doesn’t work at all.
There is no QCD model for the nuclear force
See the Wikipedia gluon article. Gluons are said to carry color charge and participate in strong interactions. Because of the color force that gluons are said to exert, quarks are said to be confined. Gluons “also share this property of being confined within hadrons” and therefore “gluons are not directly involved in the nuclear forces between hadrons”. And yet quarks and gluons are able to pop into existence and depart a proton in the guise of a 140 MeV pion? Said pion dutifully popping out of existence when it reaches the neutron? The color force is said to be always attractive and remain constant with distance, but a residue of this force is said to be repulsive at very short range, attractive at an intermediate range, and non-existent beyond 2.5 fm? Meanwhile we’ve never ever seen a quark, and the gluons in ordinary hadrons are virtual. There are no actual gluons flying back and forth inside a proton. Just as there are no actual photons flying back and forth between the protons and electron in the hydrogen molecular ion. Just as there are no actual pions flying back and forth in the nucleus. I’m afraid that’s just fairy tales and lies to children. There is no QCD model for the nuclear force. As Richard Feynman said 1985: “we have a simple definite theory which is supposed to explain all the properties of protons and neutrons yet we can’t calculate anything with it because the mathematics is too hard for us”.
The foundations of nuclear physics appear distinctly unsound
In 1986 in Hideki Yukawa and the meson theory Laurie Brown said “today’s standard model has not been able to calculate ‘low-energy’ processes, such as meson-nucleon scattering, or the nuclear forces”. In 1999 Charlotte Elster said calculations started about 15 years ago and many groups have been involved, but all the models create either too little or no intermediate-range attraction. According to Riken in 2007 the short-range repulsion remains an open question. That’s when Frank Wilczek said this in Nature: “ironically from the perspective of QCD, the foundation of nuclear physics appear distinctly unsound”. In John Gowan’s 2012 paper strong force two expressions you can read that the exact origin of the strong force is not yet a completely settled matter. In Ruprecht Machleidt’s 2013 paper origin and properties of strong inter-nucleon interactions you can read that it’s been seventy years of desperate struggle. Machleidt advocates chiral effective field theory but the bottom line is that there hasn’t been much in the way of recent progress. That’s why the nuclear force is in the list of unsolved problems in physics.
There is no model for the nuclear force
So here we are 86 years after Chadwick discovered the neutron, and physicists still don’t understand the nuclear force. We could go back further: “in 1917 (in experiments reported in 1919), Rutherford proved that the hydrogen nucleus is present in other nuclei, a result usually described as the discovery of protons”. It’s 2018 today. Rutherford discovered the proton a hundred years ago, and physicists still don’t understand the nuclear force. What a disaster. A nuclear disaster. But all is not lost. There’s a clue in the phrase electron capture. The nuclear force isn’t dead. It’s like the sleeping beauty in an ivory tower surrounded by a thorny thicket. Waiting to be wakened with a kiss. From a neutron.